BSDEs driven by time-changed Lévy noises and optimal control

A Maximum Principle for Mean-Field SDEs with Time Change

“…The existence of such representation theorems depends on the noise and the information flow fixed on the probability space. It is well known that we can obtain these results for mixtures of Gaussian and Poisson type measures and in [9] it is proved for time changed Brownian and time changed Poisson random measures. See also [10] for a specific study on the structure of the doubly stochastic Poisson random noises.…”

“…See e.g. [9] for details. The doubly stochastic noises are set in relationship with time change by the characterisation [16, Theorem 3.1] (see also [11]).…”

“…where the integrand φ can be explicitly expressed in terms of the non-anticipating derivative. See [8, Definition 3.4, Theorem 3.1] (see also [9,Theorem 3.3]).…”

“…Before entering the core of the issue we present the necessary results related to mean-field BSDEs. We follow the approach of [4] and exploit the techniques suggested in [9] and [10] for time changed Lévy noises.…”

Mean-field anticipated BSDEs driven by time-changed Lévy noises

“…Frei [6] established a new notion of local solution ofq BSDEs and quoted the implications in a financial context. Based on the work of Ren and Mohamed [16], Kallsen and Muhle-Karbe [8], Di Nunno and Sjursen (2014) [3] studied a kind of BSDEs when the noises are timechanging Lévy noises and the generator f (•) is linear. In the article they also gave the meanvariance portfolio selection.…”